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tour_groups.rst

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有限群，阿贝尔群

Sage支持排列群，有限典型群(如 SU(n,q))，有限矩阵群 （使用你的生成元） 和阿贝尔群（甚至是无限的）。很多是通过调用GAP实现的。

例如，要建立一个排列群，只需给定生成元的列表，象下面这样。

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
sage: G.order()
120
sage: G.is_abelian()
False
sage: G.derived_series()           # random-ish output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
 Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]
sage: G.center()
Permutation Group with generators [()]
sage: G.random_element()           # random output
(1,5,3)(2,4)
sage: print latex(G)
\langle (3,4), (1,2,3)(4,5) \rangle

你还可以得到特征表(LaTeX格式的):

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: latex(G.character_table())
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -\zeta_{3} - 1 & \zeta_{3} \\
1 & 1 & \zeta_{3} & -\zeta_{3} - 1 \\
3 & -1 & 0 & 0
\end{array}\right)

Sage还包括在有限域上的典型群和矩阵群：

sage: MS = MatrixSpace(GF(7), 2)
sage: gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.conjugacy_class_representatives()
    [
    [1 0]
    [0 1],
    [0 1]
    [6 1],
    ...
    [6 0]
    [0 6]
    ]
sage: G = Sp(4,GF(7))
sage: G._gap_init_()
'Sp(4, 7)'
sage: G
Symplectic Group of rank 2 over Finite Field of size 7
sage: G.random_element()             # random output
[5 5 5 1]
[0 2 6 3]
[5 0 1 0]
[4 6 3 4]
sage: G.order()
276595200

你还可以使用阿贝尔群进行计算（无限的和有限的）：

sage: F = AbelianGroup(5, [5,5,7,8,9], names='abcde')
sage: (a, b, c, d, e) = F.gens()
sage: d * b**2 * c**3
b^2*c^3*d
sage: F = AbelianGroup(3,[2]*3); F
Multiplicative Abelian Group isomorphic to C2 x C2 x C2
sage: H = AbelianGroup([2,3], names="xy"); H
Multiplicative Abelian Group isomorphic to C2 x C3
sage: AbelianGroup(5)
Multiplicative Abelian Group isomorphic to Z x Z x Z x Z x Z
sage: AbelianGroup(5).order()
+Infinity